# Symmetric difference

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## Definition

Let [ilmath]A,B\in\mathcal{P}(X)[/ilmath] be two subsets of a set [ilmath]X[/ilmath]. We define the symmetric difference of [ilmath]A[/ilmath] and [ilmath]B[/ilmath] as[1]:

• [ilmath]A\triangle B:=(A-B)\cup(B-A)[/ilmath][Note 1]
• In words: [ilmath]A\triangle B[/ilmath] contains (everything in [ilmath]A[/ilmath] and not in [ilmath]B[/ilmath]) and (everything in [ilmath]B[/ilmath] but not in [ilmath]A[/ilmath]).

Claim 1: this is equivalent to [ilmath]A\triangle B:=(A\cap B^C)\cup(A^C\cap B)[/ilmath][1]

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## Notes

1. Here [ilmath]A-B[/ilmath] denotes set subtraction.